原始范数提取与三次费马方程的带符号下降
Primitive norm extraction and a signed descent for the cubic Fermat equation
AI总结：
本文通过带符号的欧拉下降法，在 $\mathbb{Z}[\sqrt{-3}]$ 中直接提取原始范数，证明三次费马方程，并给出高度严格缩小的构造（$|C'|\le |C|/39$）。
AI中文摘要：
我们通过欧拉下降法的一种带符号形式，给出了费马定理三次情形的自包含证明，其中所需的提取步骤直接在 $\mathbb{Z}[\sqrt{-3}]$ 中证明。Thue 的鸽巢原理论证将相关素数表示为 $X^2+3Y^2$，而 $\mathbb{Z}[\sqrt{-3}]$ 中的显式除法将奇数次幂的原始范数提升为该环中的幂。对于立方情形，所得的坐标恒等式将 $2K(K^2+3M^2)=C^3$ 转化为同一带符号形式的另一个方程，且具有严格更小的非零整数高度。该构造给出 $|C'|\le |C|/39$。两个奇数项的半和与半差将和与差的奇偶配置置于这一单一下降中。
英文摘要：
We give a self-contained proof of the cubic case of Fermat's theorem through a signed form of Euler's descent, with the required extraction step proved directly in $\mathbb{Z}[\sqrt{-3}]$. A pigeonhole argument of Thue represents the relevant primes by $X^2+3Y^2$, and explicit division in $\mathbb{Z}[\sqrt{-3}]$ lifts primitive odd-power norms to powers in that order. For cubes, the resulting coordinate identities transform $2K(K^2+3M^2)=C^3$ into another equation of the same signed form with a strictly smaller non-zero integer height. The construction gives $|C'|\le |C|/39$. Half-sums and half-differences of the two odd entries place the sum and difference parity configurations in this single descent.
